Q.The demand function for a product is $x = 200 - 4p$. At $p=30$, determine whether demand is elastic, inelastic, or unit elastic, and state what would happen to total revenue if the price were increased slightly.

Find $x$ at $p=30$ and $dx/dp$ from the demand function, then apply the elasticity formula and read off the revenue effect from the classification.

$x=200-120=80$. $dx/dp=-4$. $e_d=-(30/80)(-4)=120/80=1.5$.

Step 1 — Find $x$ at $p=30$. $x = 200-4(30) = 200-120=80$.

Step 2 — Differentiate the demand function. $x=200-4p \Rightarrow \dfrac{dx}{dp}=-4$.

Step 3 — Apply the elasticity formula.

Step 4 — Classify and interpret. Since $e_d=1.5>1$, demand is elastic at $p=30$. For elastic demand, a price rise causes a proportionally larger fall in quantity demanded, so total revenue $R=xp$ decreases when price is increased (and would increase if price were instead reduced).

Independent check. Compute total revenue at $p=30$ and at a slightly higher price, say $p=31$: $R(30)=80\times30=2400$. At $p=31$: $x=200-124=76$, so $R(31)=76\times31=2356$. Since $R$ fell from $2400$ to $2356$ as price rose from $30$ to $31$, this directly confirms the elastic-demand prediction that a price increase reduces total revenue.

Assuming a price increase always raises revenue (a common intuitive but wrong assumption) — whether revenue rises or falls on a price increase depends entirely on whether demand is elastic or inelastic at that price, which is exactly why the elasticity calculation, not intuition, must decide the answer.